Properties of Gases: Comprehensive Notes

States of Matter (Review)

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• The three states of matter are solid, liquid, and gas.

Compressibility

• Compressibility refers to the change in volume of a sample due to a pressure change.

• Solids and liquids are not significantly compressible.

• Gases can be compressed under sufficient pressure.

Pressure

• Pressure is the force exerted per unit area by gas molecules striking surfaces.

• The pressure of a gas depends on:

  • Number of gas particles in a given volume

  • Volume of the container

  • Average speed of the gas particles

• Fewer gas particles result in lower pressure; higher density results in higher pressure.

Pressure Units

• $1 \text{ atm} = 760 \text{ mm Hg} = 760 \text{ torr} = 1.01325 \times 10^5 \text{ Pa} = 101.325 \text{ kPa} = 1.013 \text{ bar}$

• Average air pressure at sea level:

  • Atmosphere (atm): $1.00 \text{ atm}$

  • Inches of Mercury (in Hg): $29.921 \text{ in Hg}$

  • Torr: $760 \text{ torr}$

  • Millimeter Hg (mmHg): $760 \text{ mmHg}$

  • Pascal (Pa): $101,325 \text{ Pa}$

  • Pounds per square inch (psi): $14.7 \text{ psi}$

  • Bar: $1.013 \text{ atm}$

Barometers

• Barometers measure atmospheric pressure using the height of a mercury column.

Atmospheric Pressure Example

• Heating water in a sealed can, then cooling it, causes the can to crumple due to external atmospheric pressure being higher than the internal pressure.

Manometers

Closed-end Manometers:

• Used to measure gas pressure

• $P_{\text{gas}} = \Delta h$

• If P{\text{gas}} << P{\text{atm}}, use a closed-end manometer. The pressure of the gas in the flask is equal to the change in height of the column of liquid (e.g. Mercury).

Open-end Manometers:

• Used to measure gas pressure by comparing gas pressure to atmospheric pressure.

• $P<em>{\text{gas}} = P</em>{\text{atm}} \pm \Delta h$

• If mercury levels are equal, $P<em>{\text{gas}} = P</em>{\text{atm}}$

• If the mercury level is higher on the gas side, P{\text{gas}} < P{\text{atm}}, so $P<em>{\text{gas}} = P</em>{\text{atm}} - \Delta h$

• If the mercury level is higher on the atmosphere side, P{\text{gas}} > P{\text{atm}}, so $P<em>{\text{gas}} = P</em>{\text{atm}} + \Delta h$

Manometer Problems

1. Draw and label manometer.

2. Determine if P{\text{gas}} > P{\text{atm}} or P{\text{gas}} < P{\text{atm}}.

3. If P{\text{gas}} > P{\text{atm}}, $P<em>{\text{gas}} = P</em>{\text{atm}} + \Delta h$

4. If P{\text{gas}} < P{\text{atm}}, $P<em>{\text{gas}} = P</em>{\text{atm}} - \Delta h$

5. Always make certain your units match.

Non-Mercury Manometers

• $h<em>{\text{liquid}} = \frac{d</em>{\text{Hg}} \cdot h<em>{\text{Hg}}}{d</em>{\text{liquid}}}$

• $h<em>{\text{liquid}} \times d</em>{\text{liquid}} = h<em>{\text{Hg}} \times d</em>{\text{Hg}}$

• For a $760 \text{ mm}$ column of Hg ($d_{\text{Hg}} = 13.6 \text{ g/mL}$), the corresponding height (h) of a column of water ($d = 1.0 \text{ g/mL}$) would be $10,300 \text{ mm}$ (nearly $34 \text{ feet}$).

Gas Laws

• Gas laws are mathematical relationships that describe the quantitative behavior of gases.

• Gas properties (P, V, n, T) are interrelated—when one changes, it affects the others.

The Gas Constant, R

• R (gas constant) is the proportionality constant in the ideal gas equation.

Basic Gas Laws

• Boyle’s law: relates pressure and volume

• Charles’s law: relates volume and temperature

• Avogadro’s law: relates volume and moles of a gas

Boyle's Law

• Volume and Pressure are inversely proportional when the mol quantity of gas and temperature are constant.

• $V \propto \frac{1}{P}$

• $P<em>1 \times V</em>1 = P<em>2 \times V</em>2$

• From $PV=nRT$ when n, R, and T are constant then $P<em>1V</em>1 = P<em>2V</em>2$

Charles's Law

• Volume and absolute temperature (in K) are directly proportional when mol quantity of gas and pressure are constant.

• $V \propto T$

• From $PV=nRT$ when n, R, and P are constant then $\frac{V<em>1}{T</em>1} = \frac{V<em>2}{T</em>2}$

Avogadro's Law

• The volume of a gas is directly proportional to the number of moles of the gas when temperature and pressure are held constant.

• The volume of a gas sample increases linearly with the number of moles of gas in the sample.

• $n \propto V$

• From $PV=nRT$ when P, R, and T are constant then $\frac{V<em>1}{n</em>1} = \frac{V<em>2}{n</em>2}$

• Equal volumes of gases contain equal numbers of molecules at the same T, P (the gas doesn't matter).

• Two samples of gas with the same volume, pressure, and temperature will have the same number of moles of gas present.

• $\frac{n<em>1}{V</em>1} = \frac{n<em>2}{V</em>2}$

Using PV = nRT

When variables Change:

• n, P, T, V: some change, some may be constant

• Units must match each other.

• Note: When some of the variables remain constant, you do not have to include them in the calculation. Thus using Boyle’s Law, Charles’s Law. Etc.
  $\frac{P<em>1V</em>1}{n<em>1T</em>1} = \frac{P<em>2V</em>2}{n<em>2T</em>2}$

No change in variables:

• PV = nRT: one value for each, no change

• Units must match the R units.

• Note: solving for n leads to calculating mass in g.

• $PV = nRT$

STP

• STP stands for Standard Temperature and Pressure

Density & Molar Mass

• Using $PV = nRT$ you can calculate the density of a gas:

• $d = \frac{m}{V} = \frac{P \cdot MolarMass}{R \cdot T}$

Gas Mixtures and Partial Pressure

• Gases are not always pure – mixtures of two or more gases are often of interest. In a mixture, the gases are:

  • in the same container (same V)

  • are at the same temperature (same T)

  • R is a constant, so is always the same

• $P = \frac{nRT}{V}$

• The pressure of each gas in separate containers may be calculated by:

  • $P<em>1 = \frac{n</em>1RT}{V}$

  • $P<em>2 = \frac{n</em>2RT}{V}$

  • $P<em>3 = \frac{n</em>3RT}{V}$

Gas Mixtures: Total Pressure Summary

• The Total Pressure of a mixture of gases may therefore be calculated by:

  • Calculating the individual pressure contributions by each gas and then summing them: $P<em>{\text{total}} = P</em>1 + P<em>2 + … + P</em>n$

  • Summing the mol quantities of each gas: $P<em>{\text{total}} = (n</em>1 + n<em>2 + n</em>3 + …) \frac{RT}{V}$

Gas Mixtures and Partial Pressure

• The partial pressure of each gas in a mixture of gases is the pressure contribution of only that particular gas.

• The partial pressure of a gas can be calculated by:

  • $P<em>1 = \frac{n</em>1RT}{V}$

• Only considering the mol quantity of the one gas in the container.

• The mole fraction of the mixture it composes is known along with the total pressure.

Partial Pressure and Mole Fraction

• The ratio of the moles of one gas to the total mole quantity is called the mole fraction ($\chi$).

• $\chi = \frac{n<em>{\text{gas}}}{n</em>{\text{total}}}$

• The partial pressure of a gas is determined by rearranging the above equation to yield its mole fraction multiplied by the total pressure.

• $P<em>1 = \chi</em>{\text{gas}} (P_{\text{total}})$

Collecting Gases Over Water

• A common method of gas collection in the laboratory is the collection of the gas over water.

• Gases collected over water cannot be readily soluble in water and cannot react with water.

• The collection vessel will contain both the collected gas(es) and water vapor.

• Use a table of water vapor pressures to determine the water vapor pressure (partial pressure).

Kinetic Molecular Theory

• The simplest model for the behavior of gases is the kinetic molecular theory.

• There are five postulates to the kinetic molecular theory (KMT) of gases.

Kinetic Molecular Theory: Postulates

• Postulate 1: A gas is composed of a large number of particles called molecules (whether monatomic or polyatomic) that are in constant random motion.

  • Because the gas particles are constantly moving, they strike the sides of the container with a force.

  • The result of many particles in a gas sample exerting forces on the surfaces around them is a constant pressure.

• Postulate 2: Because the distance between gas molecules is much greater than the size of the molecules, the volume of the molecules is negligible.

  • Gases are “point masses” and the size of the particle is insignificant when compared to the space between molecules.

  • The large space between gas particles explains the ability for gases to be significantly compressed.

• Postulate 3: Intermolecular interactions, whether repulsive or attractive, are so weak that they are also negligible.

  • If all gas particles behave alike, regardless of the chemical nature of their component molecules (intermolecular forces exhibited, for example), gases will follow the Ideal Gas Law.

  • $PV = nRT$

  • The ideal gas law treats all gases as collections of particles that are identical in all respects except mass.

• Postulate 4: Gas molecules collide with one another and with the walls of the container, but these collisions are perfectly elastic; that is, they do not change the average kinetic energy of the molecules.

  • This means that when two particles collide, they may exchange energy, but there is no overall loss of energy.

  • Any kinetic energy lost by one particle is completely gained by the other.

• Postulate 5: The average kinetic energy of the molecules of any gas depends on only the temperature, and at a given temperature, all gaseous molecules have exactly the same average kinetic energy.

  • The average kinetic energy of the gas particles is directly proportional to the Kelvin (absolute) temperature.

  • The temperature of the gas increases, the average speed of the particles increases.

  • Not all the gas particles are moving at the same speed.

  • $KE = \frac{1}{2}mv^2$

Kinetic Molecular Theory & Gas Laws

• Boyle’s Law: V and P are inversely proportional (n, T constant).

  • Decreasing the volume forces the molecules into a smaller space.

  • More molecules will collide with the container at any one instant, resulting in an increase in pressure.

• Charles’s Law: V and T are directly proportional to the absolute temperature (P, n constant)

  • Volume must increase to allow pressure to remain constant.

  • Increasing the temp (K) of a gas makes the gas particles move faster, therefore increasing its average kinetic energy.

• Pressure and Temperature: At constant volume, pressure is proportional to absolute temperature.

  • When temp increases, the pressure increases if the volume does not change.

  • Particles will strike the container more frequently in the same volume.

• Avogadro’s Law: Volume is directly proportional to the number of gas molecules.

  • Increasing the number of gas molecules results in more collisions with the container walls.

  • To keep the pressure constant, the volume must then increase.

• Dalton’s Law of Partial Pressure

  • According to kinetic molecular theory, the particles have negligible size and they do not interact.

  • Particles of different masses have the same average kinetic energy at a given temperature.

  • Because the average kinetic energy is the same, the total pressure of the collisions is the same.

Root-Mean-Square Speed

• $u_{rms} = \sqrt{\frac{3RT}{M}}$

• The kinetic energy of gas molecules is related to the root-mean-square speed ($u_{rms}$) of a gas.

• The root-mean-square speed ($u_{rms}$) is the square root of the average of the squared speeds of the gas molecules in a gas sample.

• Temperature is in the numerator: increase T, increase $u_{rms}$.

• Molar Mass is in the denominator: increase M, decrease $u_{rms}$.

Temperature and Molecular Velocities

• At a given temperature:

  • All ideal gases have the same average kinetic energy but the velocity of individual gases depends on molar mass.

  • Heavier molecules move more slowly than lighter ones.

Gases in Chemical Reactions: Stoichiometry Revisited

• For stoichiometric calculations involving gases, we can use the ideal gas law to determine the amounts in moles from the volumes of gases.

• $PV = nRT$ therefore $n = \frac{PV}{RT}$

Real Gases: Deviations from Ideal Behavior

• In summary, real gases will behave more like ideal gases under conditions of:

  • Low pressure (at higher pressures, the volume of the gas molecules becomes significant)

  • High temperatures (at lower temperatures, the intermolecular attractions can lower the expected pressures)

• According to the Kinetic Molecular Theory, ideal gas laws assume:

  • no attractions between gas molecules.

  • gas molecules do not take up space.

• At low temperatures and high pressures these assumptions are not valid.

• When compressed, real gas particles occupy a more significant portion of the total gas volume.

• Because real molecules take up space, the molar volume of a real gas is larger than predicted by the ideal gas law at high pressures.

• The behavior of real gases is close to that of ideal gases at low pressures (generally less than 10 atm).

• Greater deviation from ideal gases is noted as the temperature decreases.

Real Gases: Temperature Effects

• At lower temperatures, the kinetic energy of the molecules will be lower.

• The attractions between the molecules are the same, however.

• Reducing the temperature (thus the kinetic energy) allows molecules to experience greater intermolecular attractions, since the energy of the molecules will be insufficient to overcome the attractive forces.

• At higher temperatures, molecules have greater energy to overcome these attractive forces.

Van der Waals Equation

• In order to predict the behavior of real gases more accurately, Dutch scientist Johannes van der Waals developed a mathematical equation.

• van der Waals equation: An equation of state for nonideal gases that is based on adding corrections to the ideal gas equation.

• van der Waals equation corrections account for:

  • Intermolecular forces of attraction

  • Volume occupied by gas molecules

Real Gases: Van der Waals Equation

• $\left(P + a\frac{n^2}{V^2}\right)(V - nb) = nRT$

• b is a measure of the volume occupied by a mole of gas molecules

• a reflects the strength of the attraction of the gas particles.

• At STP, adjustments will be near zero. Therefore, PV = nRT